Abstract

In this paper, we introduce the concept of new notions related to n-tupled fixed point and prove some related results for an asymptotically regular one-parameter semigroup ℑ={F(t):t∈G,where G is an unbounded subset of [0,∞)} of Lipschitzian self-mappings on ∏i=1nX in the case when (X,d) is a complete bounded metric space with uniform normal structure. Our results extend the results due to Yao and Zeng (J. Nonlinear Convex Anal. 8(1):153-163, 2007) and Soliman (Fixed Point Theory Appl. 2013:346, 2013; J. Adv. Math. Stud. 7(2):2-14, 2014).

Keywords

1 Introduction

The Banach contraction principle is the most natural and significant result of fixed point theory. In complete metric spaces it continues to be an indispensable and effective tool in theory and applications, which guarantees the existence and uniqueness of fixed points of contraction self-mappings besides offering a constructive procedure to compute the fixed point of the underlying mapping. There already exists an extensive literature on this topic. Keeping in view the relevance of this paper, we merely refer to [1–5]. In 1987, the idea of coupled fixed point was initiated by Guo and Lakshmikantham [6]; it was also followed by Bhaskar and Lakshmikantham [7] wherein authors proved some interesting coupled fixed point theorems for mappings satisfying the mixed monotone property. Many authors obtained important coupled, tripled and n-tupled fixed point theorems (see [7–16]). In this continuation, Lakshmikantham and Ćirić [13] introduced coupled common fixed point theorems for nonlinear ϕ-contraction mappings in partially ordered complete metric spaces which indeed generalize the corresponding fixed point theorems contained in Bhaskar and Lakshmikantham [7]. In 2010, Samet and Vetro [17] introduced the concept of fixed point of n-tupled fixed point (where n=2,3,4,…) for nonlinear mappings in complete metric spaces. They obtained the existence and uniqueness theorems for contractive type mappings. Their results generalized and extended coupled fixed point theorems established by Bhaskar and Lakshmikantham [7]. Recently, Imdad et al. [18] introduced a generalization of n-tupled fixed point and n-tupled coincidence point by considering n even besides using the idea of mixed g-monotone property on ∏i=1nX and proved an n-tupled (where n is even) coincidence point theorem for nonlinear ϕ-contraction mappings satisfying the mixed g-monotone property. For more information about n-tupled fixed points, see [10, 17–19].

On the other hand, normal structure is one of the most important aspects of metric fixed point theory. It was introduced by Brodskii and Milman in [20]. They found the first application of normal structure to fixed point theory. In 1965, Kirk [21] introduced the following theorem: Every nonexpansive self-mapping on a weakly compact convex subset of a Banach space with normal structure has a fixed point. In 1969, Kijima and Takahashi [22] established the metric space version of Kirk’s theorem [21]. Subsequently, many authors successfully generalized certain fixed point theorems and structure properties from Banach spaces to metric spaces. For example, Khamsi [23] defined normal and uniform normal structure for metric spaces and proved that if (X,d) is a complete bounded metric space with uniform normal structure, then it has the fixed point property for nonexpansive mappings and a kind of intersection property which extends a result of Maluta [24] to metric spaces. In 1995, Lim and Xu [25] proved a fixed point theorem for uniformly Lipschitzian mappings in metric spaces with both property (P) and uniform normal structure, which extended the result of Khamsi [23]. This is the metric space version of Casini and Maluta’s theorem [2]. In 2007, Yao and Zeng [26] established a fixed point theorem for an asymptotically regular one-parameter semigroup of uniformly k-Lipschitzian mappings with property (∗) in a complete bounded metric space with uniform normal structure, which extended the results of Lim and Xu [25]. Recently, the idea of coupled and tripled fixed point results in a complete bounded metric space X with uniform normal structure was initiated by Soliman [27, 28]. He proved that every asymptotically regular one-parameter semigroup ℑ={F(t):t∈G,} of Lipschitzian mappings on X×X has a coupled fixed point and on X×X×X has a tripled fixed point.

In the present paper, we prove an n-tupled fixed point theorem for asymptotically regular Lipschitzian one-parameter semigroups ℑ={F(t):t∈G} on ∏i=1nX, where X is a complete bounded metric space with uniform normal structure. Also, some corollaries of our main theorem are presented.

2 Preliminaries

Let (X,≤) be a partially ordered set and F:X×X→X. We say that F has the mixed monotone property if F(x,y) is monotone nondecreasing in x and is monotone nonincreasing in y, that is, for any x,y∈X, x1,x2∈X, x1≤x2⇒F(x1,y)≤F(x2,y) and y1,y2∈X, y1≤y2⇒F(x,y1)≥F(x,y2).

Let(X,≤)be a partially ordered set and suppose that there is a metricdonXsuch that(X,d)is a complete metric space. LetF:X×X→Xbe a continuous mapping having the mixed monotone property onX. Assume that there exists a constantk∈[0,1)with

d(F(x,y),F(u,v))≤k2[d(x,u)+d(y,v)]∀x≥u,y≤v.

If there existx0,y0∈Xsuch thatx0≤F(x0,y0)andy0≥F(y0,x0), then there existx,y∈Xsuch thatx=F(x,y)andy=F(y,x).

A metric space (X,d) is said to have normal (resp. uniform normal) structure if there exists a convexity structure μ on X such that R(A)<δ(A) (resp. R(A)≤c⋅δ(A) for some constant c∈(0,1)) for all A∈μ which is bounded and consists of more than one point. In this case μ is said to be normal (resp. uniformly normal) in X.

We define the normal structure coefficient N(X) of X (with respect to a given convexity structure μ) as the number

sup{R(A)δ(A)},

where the supremum is taken over all bounded A∈F with δ(A)>0. X then has uniform normal structure if and only if N(X)<1.

Khamsi proved the following result that will be very useful in the proof of our main theorem.

A metric space (X,d) is said to have property (P) if given any two bounded sequences {xn} and {zn} in X, one can find some z∈⋂n=1∞ad{zj:j≥n} such that

lim supn→∞d(z,xn)≤lim supj→∞lim supn→∞d(zj,xn).

3 Main results

Let G be a subsemigroup of [0,∞) with addition ‘+’ such that

t−s∈G∀t,s∈G with t≥s.

This condition is satisfied if G=[0,∞) or G=Z+, the set of nonnegative integers. Let ℑ={F(t):t∈G} be a family of self-mappings on ∏i=1rX. Then ℑ is called a (one-parameter) semigroup on ∏i=1rX if the following conditions are satisfied:

Definition 3.4 Let (X,d) be a complete bounded metric space and ℑ={F(t):t∈G} be a semigroup on ∏i=1rX. Then ℑ has property (∗) if for each x∈X and each {tn}∈w(∞), the following conditions are satisfied:

Remark 3.1 If X is a complete bounded metric space with property (P), then each semigroup ℑ={F(t):t∈G} on ∏i=1rX has property (∗).

Lemma 3.1Let(X,d)be a complete bounded metric space with uniform normal structure, and letℑ={F(t):t∈G}be a semigroup on∏i=1rXwith property (∗). Then, for eachx∈X, each{tn}∈ω(∞)and for any constantN˜(X)<c, the normal structure coefficient with respect to the given convexity structureμ, there exist somea1∈⋂n=1∞ad{aj1:j≥n}, …, ar∈⋂n=1∞ad{ajr:j≥n}satisfying the following properties:

Proof For each integer n≥1, let An1={F(tj)(x1,x2,x3,…,xr):j≥n}, An2={F(tj)(x2,x3,x4,…,xr,x1):j≥n}, …, Anr={F(tj)(xr,x1,x2,…,xr−1):j≥n}. Then {An1},{An2},…,{Anr} are decreasing sequences of admissible subsets of X hence A1:=⋂n=1∞An1≠ϕ, A2:=⋂n=1∞An2≠ϕ, …, Ar:=⋂n=1∞Anr≠ϕ by Proposition 2.2. From Proposition 2.1, it is not difficult to see that δ(An1)=δ({F(ti)(x1,x2,x3,…,xr):i≥n}), δ(An2)=δ({F(ti)(x2,x3,x4,…,xr,x1):i≥n}), …, δ(Anr)=δ({F(ti)(xr,x1,x2,…,xr−1):i≥n}). Indeed, observe that

We now suppose that for each n≥1, there exist an1∈An1, an2∈An2, …, anr∈Anr such that

r(an1,An1)≤c⋅δ({F(tj)(x1,x2,x3,…,xr):j≥n}),

(3)

r(an2,An2)≤c⋅δ({F(tj)(x2,x3,x4,…,xr,x1):j≥n}),

(4)

⋮r(an2,Anr)≤c⋅δ({F(tj)(xr,x1,x2,…,xr−1):j≥n}).

(5)

Indeed, if δ({F(tj)(x1,x2,x3,…,xr):j≥n})=0, then δ(An1)=δ({F(tj)(x1,x2,x3,…,xr):j≥n}), we conclude that (3) holds. Without loss of generality, we may assume that δ({F(tj)(x1,x2,x3,…,xr):j≥0})>0. Then, for N(X)<c, we choose ϵ>0 so small that it satisfies the following:

Thus, we have ∑l=0∞d(xl+11,xl1)≤2D0max{h,kr}∑l=0∞hl−1<∞, …, ∑l=0∞d(xl+1r,xlr)<∞. Consequently, {xl1},…,{xlr} are Cauchy and hence convergent as X is complete. Let x1=liml→∞xl1, …, xr=liml→∞xlr, then, for each s∈G, by the continuity of F(s) we have

Remark 3.2 It is well known that the Lipschitzian mapping is uniformly continuous. It is natural to ask if there is a contractive mapping definition which does not force it to be continuous. It was answered affirmatively by Kannan. It is clear that Lipschitzian mappings are always continuous and Kannan type mappings are not necessarily continuous. It will be interesting to establish Theorem 3.1 for representative ψ={F(t):t∈G} on ∏i=1rX satisfying the following condition:

Copyright

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